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cos(npi)/5^n

Sum of series cos(npi)/5^n



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The solution

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  oo           
____           
\   `          
 \    cos(n*pi)
  \   ---------
  /        n   
 /        5    
/___,          
n = 1          
$$\sum_{n=1}^{\infty} \frac{\cos{\left(\pi n \right)}}{5^{n}}$$
Sum(cos(n*pi)/5^n, (n, 1, oo))
The radius of convergence of the power series
Given number:
$$\frac{\cos{\left(\pi n \right)}}{5^{n}}$$
It is a series of species
$$a_{n} \left(c x - x_{0}\right)^{d n}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}$$
In this case
$$a_{n} = \cos{\left(\pi n \right)}$$
and
$$x_{0} = -5$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-5 + \lim_{n \to \infty} \left|{\frac{\cos{\left(\pi n \right)}}{\cos{\left(\pi \left(n + 1\right) \right)}}}\right|\right)$$
Let's take the limit
we find
$$\frac{1}{R} = \tilde{\infty} \left(-5 + \lim_{n \to \infty} \left|{\frac{\cos{\left(\pi n \right)}}{\cos{\left(\pi \left(n + 1\right) \right)}}}\right|\right)$$
$$R = 0 \left(-5 + \lim_{n \to \infty} \left|{\frac{\cos{\left(\pi n \right)}}{\cos{\left(\pi \left(n + 1\right) \right)}}}\right|\right)^{-1}$$
The rate of convergence of the power series
The answer [src]
  oo               
 ___               
 \  `              
  \    -n          
  /   5  *cos(pi*n)
 /__,              
n = 1              
$$\sum_{n=1}^{\infty} 5^{- n} \cos{\left(\pi n \right)}$$
Sum(5^(-n)*cos(pi*n), (n, 1, oo))
Numerical answer [src]
-0.166666666666666666666666666667
-0.166666666666666666666666666667
The graph
Sum of series cos(npi)/5^n

    Examples of finding the sum of a series